• ### General

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1 Introduction to electrodynamics: Maxwell equations, vector and scalar potentials, presentation of the electromagnetic field as a superposition of the plane waves, electromagnetic field energy.
Landau & Lifshitz, § 52
2 Canonical variables, electromagnetic field as a superposition of harmonic oscillators, quantization of the field, photons. Thermal (black) radiation, Plank equation, Einstein theory of spontaneous and stimulated emission. Landau & Lifshitz, § 52
Demtroder, 2.2-2.3
Corney, 9.2
3 Spontaneous emission, expression for the spontaneous emission probability. Oscillator strength. Introduction to line broadening. Natural broadening. Corney, 4.1-4.3, 4.6, 4.8, 8, 8.1.
4 Pressure (collisional) broadening for the impact approximations. Pressure shift.  Corney, 8.2 (excluding 8.2.4 b and c)
Demtroder, 3.3 (pressure shift)
5  Doppler broadening. Voigt function, transit-time broadening.  Corney, 8.3-8.5
Demtroder, 3.4 (transit-time broadening)
6 Radiation transfer. Beer-Lambert law and expression for the absorption coefficient. Optically thick and thin layers. Self-reversal of the line. Corney, 10.1-10.3
7 Rate equations. Three and four level systems. Conditions for population inversion in both systems. Siegman, 6.1
8 Gain saturation for the cases of Doppler and Lorentz broadening. Hole burning. Absorption spectroscopy in gases.
Yariv,  5.6-5.7
Corney, 13.3 -13.4

9Laser amplification. Amplifier as a filter: narrowing of the line in the amplifier. Amplification of the strong signal: homogeneous and inhomogeneous broadening.
Siegman, 7.7
10 Laser oscillation. Self-excitation condition. Single and multimode oscillation for homogeneous and inhomogeneous saturation, respectively. Optical resonator modes, width of the mode, Q-factor. Yariv, 6.0-6.1
Siegman, 12.1
11 Power extraction in the case of homogeneous broadening. Small output coupling and constant intraresonator intensity approximation. Optimal mirror transmission and optimal power. Siegman, 12.3
Yariv, 6.4-6.5
12 Rigrod model for the laser power in the case of the arbitrary output coupling. Q-switching.
Rigrod:
Siegman, 12.4
Q-switching:
Siegman, 26.1-2
Yariv, 6.7
13 Paraxial wave equation, Gaussian beams, stable optical resonators. Siegman, 16.1-16.3, 19.1-19.2

• There is no one main textbook in the course. Some of the books, which are relevant to the course are listed bellow. The relevant chapters could be read from the table above.

• Corney, Atomic and laser spectroscopy
• Siegman, Lasers
• Yariv, Introduction to optical electronics
• Demtroder, Laser spectroscopy
• Landau & Lifshitz, The Classical Theory of Fields

• #### Office hours

Name Day Hours Building/Room E-mail
Prof. Boris Barmashenko By appointment - 54/323 barmash@bgu.ac.il

#### Lecture/Tutorial

Group What? Name Day Hours Building/Room
1 Lecture Prof. Boris Barmashenko
Thursday9-12 35/213

• 1.      Derive Plank’s equation for the energy density of thermal radiation.

2.      Derive Wien displacement lаw: show that lm corresponding to the maximum intensity of radiation is given by λm≈0.2hc/kT.

3.      Using phenomenological approach of Einstein find relations between the coefficients of absorption (Blu), spontaneous (Alu) and stimulated (Bul) emission.

4.      Using the model of damped harmonic oscillator find the natural line shape.

5.      Show that in the case of collisional broadening in the impact approximation the line profile is Lorentzian (Lorentz theory) (Corney)

6.      Derive an expression for the line profile in the case of Doppler broadening.

7.      Derive an expression for the line profile in the gas if the Doppler and Lorentz line-widths are of the same order of magnitude (Voigt function). (Doppler line profile is given by G(w)  = (4 ln2/πΔωD)1/2exp[-4ln2(ω-ω0)2/ΔωD2].

8.   Show that in the line center (ω = ω0) the value of the Voigt function is smaller than the value of the Lorenz line shape

and that in the wings of the line (ω - ω0>>ΔωD, ΔωL) the line profile is always Lorentzian.

9.   Derive an expression for absorption coefficient k in terms of the wavelength l, the spontaneous coefficient A, lineshape functiom g(ν) and the number densities of the particles in the lower and upper level, Nl and Nu, respectively.

10.  Derive an expression for the flux of radiation energy of frequency w emitted by the layer of depth l. Assume that the layer consists of the two-level molecules with the energy of transition hω.  The temperature is T. Consider only the limiting cases of the optically thick and thin layer.

11.  Estimate the linewidth of the radiation emitted by the optically thick body in the case of Lorentzian broadening. The product of the absorption coefficient in the center of line k0 and the dimension of the body l is much greater than unity.

12.  Find dependencies of the population inversion on the pumping rate for 4- and 3- level systems using rate equations. For what system (3- or 4- level) is it easier to get population inversion and why?

13.  Derive the homogeneous saturation law: find the gain at frequency $$\omega$$, when a strong saturating signal of intensity I is applied at the same frequency w in the case of homogeneous (Lorentzian) broadening. Assume that the small signal gain in the line center (frequency ω0) is g0, the life time of the upper level $$\tau$$ and stimulated emission cross section $$\sigma$$.

14.  Derive the inhomogeneous saturation law: find the gain at frequency $$\omega$$ when a strong saturating signal of intensity I is applied at the same frequency $$\omega$$ in the case of homogeneous (Doppler) broadening (ΔωD>>ΔωL). Assume that the small signal gain in the line center (frequency ω0) is g0 and the saturation parameter is Is,res.

15.  Line narrowing in the amplifier. Find the linewidth after passing through the laser amplifier with the unsaturated small signal gain α0 and length l; assume that before the amplifier the line has a uniform spectral intensity profile and that homogeneous (Lorentz) broadening takes place. Neglect saturation effects.

16.  Saturation in laser amplifier. Laser light with intensity Iin is passing through the amplifier with unsaturated small signal gain α0 and length l.

i) Find the output intensity Iout assuming homogeneous broadening with saturation parameter IS. Find maximum available power which can be extracted from the amplifier; for which Iin this power can be extracted. Assume that the frequency of the amplified beam is equal to the central frequency of the laser transition.

ii) The same for in-homogeneous broadening in the gain medium.

17.  Laser oscillation. Find the output power of the laser with small signal gain α0 and the gain length l. The transmission of the output mirror is t, the absorption/scattering losses of this mirror are equal to a. Find optimal transmission of the mirror, assume that intensity is uniform inside the resonator.

18.  Q-switching. Q-switching is performed in the laser with the initial population density Ni of the upper laser level to get a laser pulse. The length of the laser resonator is l and the total mirror transmission is T. The stimulated emission cross section on the laser transition is σ.

i) Find the threshold population of the upper laser level Nth such that the Q-switching is possible only for Ni> Nth. Assume that the 4-level scheme is employed so that the lower laser level is empty.

ii) Derive the relation between the photon density n and the population N of the upper laser level for any moment t.

iii) Find the maximum laser power in the pulse.

• ##### Lectures

The lectures will be held in hybrid mode